Optimal Card Guessing When You Count What Is Gone

Problem. A dealer turns over a shuffled 52-card deck one card at a time. Before each card is revealed, you guess its suit. You get to see every card as it's revealed (so you know exactly what's left). What's the best strategy, and how many correct guesses do you expect over the whole deck — compared to a player who guesses blind every time?

The blind baseline

If you ignore all information and just guess the same suit every time (or guess randomly among the 4 suits each time), your chance of being right on any single card is always $1/4$, so your expected correct guesses over 52 cards is $52 \times \frac{1}{4} = 13$. This is the benchmark you're trying to beat — and the whole point of the question is that you can beat it, because the deck is finite and shrinking, so what's left becomes more predictable as the deal proceeds.

The optimal strategy: guess the majority suit remaining

At any point, some suits have more cards left in the shrinking deck than others (unless it's an exact tie). The best strategy is always to guess whichever suit currently has the most cards remaining — this maximizes your probability of being correct on that specific card, since the chance of the next card being suit $s$ is exactly (cards of suit $s$ left) / (cards left total), and picking the largest such fraction beats picking any other suit or guessing randomly.

Worked example: why counting beats guessing blind

Consider the very last card in the deck. You've seen all 51 others, so you know with certainty what suit the final card is — your probability of being correct is $100\%$, not $25\%$. Now consider the second-to-last card: two cards remain, and you know their two suits (possibly the same suit twice, possibly two different ones). If they're the same suit, you're guaranteed correct; if different, you're right with probability at least $1/2$ — never worse than blind guessing, and often much better. This pattern holds throughout the deck: as cards are revealed, the remaining suit composition drifts away from an even $13/13/13/13$ split just by chance, and always guessing the current majority suit captures that drift.

The expected total, worked out by summing the "always guess the max-remaining-suit" probability over all 52 positions, comes to roughly $13 + 4\left(\ln 13 + \gamma\right)$-ish territory in the classical analysis of this problem (the exact constant isn't something you're expected to derive live) — landing meaningfully above 13, generally cited around 17–18 correct guesses out of 52 with optimal counting, versus 13 for blind guessing. The exact number matters less than the mechanism: information about what's already gone makes the future more predictable, and the gain compounds as the deck empties, since the last few cards are nearly certain.